Journal of Glaciolog)', Vol. 27, No. 95, 1981

THE STEADY PROFILE OF AN AXISYMMETRIC ICE SHEET

By I. R. ] OHNSON

(School of Mathematics and Physics, University of East Anglia, Norwich NR4 7T], England)

ABSTRACT. Steady plane flow under gravity of an axisymmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding is treated according to a power law between shear traction and velocity. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, and temperature variation through the ice sheet is neglected. Illustrations are presented for Glen's power law (including the special case of a Newtonian fluid), and the polynomial law of Colbeck and Evans. The analysis follows that of Morland and Johnson ( lg80) where the analogous problem for an ice sheet deforming under plane flow was considered. Comparisons are made between the two models and it is found that the effect of the third dimension is to reduce (or leave unchanged) the aspect ratio for the cases considered, although no general formula can be obtained. This reduction is seen to depend on both the surface accumulation and the sliding law.

RESUME. Profit d' i quitibre d'une catotle gtaciaire ayant un axe de symitrie. On traite de l'ecoulement plan permanent sous l'influence de la gravite d'une calotte glaciaire presentant un axe de symetrie reposant sur un lit rigide horizontal, soumis en surface a une accumulation et une ablation, une evacuation au fond et un glissement sur le lit suivant une loi-puissance pour la relation vitesse/cisaillement. L'accumulation en surface a ete choisie comme liee a la hauteur, les coefficients d'ecoulement et de glissement dependent egalement de la hauteur de la glace susjacente. La glace est consideree comme un fluide incompressible non lineairement visqueux et on neglige la variation de la temperature a l'interieur de la glace. On prescrit des illustrations correspond ant a la loi-puissance de Glen (y compris le cas particulier du fluide Newtonien) et a la loi poly­nomiale de Col beck et Evans. L'analyse fait suite a ce!le de Morland et Johnson ( lg80) ou un probleme analogue a ete aborde pour une calotte glaciaire se deformant par un ecoulement plan. On fait des com­paraisons entre les deux modeles et on trouve que l'effet de la troisieme dimension est de reduire (ou de laisser inchange) le rapport de relief dans les cas consideres bien qu'on ne puisse pas obtenir de formule generale. On considere que cette reduction depend a la fois de I'accumulation en surface et de la loi de glissement.

ZUSAMMENFASSUNG. Das stationare Profit eines achssymetrisclien Eisschildes. Es wird der stationare, ebene Fluss unter Schwerkraft eines achssymetrischen Eisschildes, der auf einem horizontalen, starren Untergrund ruht und sowohl Akkumulation und Ablation an der Oberfiach e, als auch Abfluss und Gleiten am Untergrund erfahrt, nach einem Potenzgesetz fur die Geschwindigkeit bei Scherung und Zug behandelt. Fur die Akkumulation an der Oberflache wird Abhangigkeit von der Hiihe angenommen; die Koeffizienten fur Abfluss und Gleiten hangen von der Hiihe des uberlagernden Eises ab. Das Eis wird als eine allgemeine, nicht-linear viskose, unkomprimierbare Fltissigkeit beschrieben; Temperaturschwankungen im Eisschild werden vernachlassigt. Darstellungen beziehen sich auf Glen's Potenzgesetzt (einschliesslich des Sonderfalls einer Newton'schen Flussigkeit) und auf das Polynomgesetz von Col beck und Evans. Die Analyse folgt der von Morland undJohnson ( lg80), wo das analogue Problem fur eincn Eisschild, der sich unter ebenem Fluss verformt, betrachtet wurde. Zwischen den beiden Modellen werden Vergleiche gezogen, wobei si ch feststellen lasst, dass die Einbeziehung der dritten Dimension eine R eduktion (bzw. keine Veranderung) des Umriss­verhaltnisses der beiden betrachteten Falle zur Folge hat, obwohl keine allgemeine Forme! zu gewinnen ist. Diese Reduktion erweist sich als abhangig sowohl von der Oberflachenakkumulation wie vom Gleitgesetz.

I. INTRODUCTION

The mechanics of a bounded ice sheet with steady free surface has been considered by Nye (1959), Weertman (1961), and more recently by Morland and ] ohnson (1980). Nye considered both plane and axisymmetric flow in order to estimate the effects of a third dimension in comparison with the plane-flow restriction. Morland and ]ohnson's analysis was for an ice sheet deforming in plane flow only, solving the full momentum equations by a regular perturbation technique. The equivalent axisymmetric solution is now presented. There are large similarities between the plane and axisymmetric analyses and details which apply to both problems are not repeated. Henceforth Morland and ]ohnson (1980) will be referred to as M and J.

Consider an ice sheet shown by the cross-section in Figure I, with horizontal bed z = ° and surface z = h(r) in cylindrical polar coordinates (r, 8, z), and all physical variables independent of 8. The velocity components are (u, 0, w) and g is the constant acceleration due to gravity. 't is the Cauchy stress tensor with cry(} = crrz = 0, and (tn, ts) denote normal and tangential tractions on the surface. The ice sheet is maintained in steady flow by surface accumulation and ablation and basal drainage, with zero net flux.

25

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z

Fig. 1. Ice-sheet cross-section.

By considering constant surface accumulation Nye (1959) obtained the solution

(~)HI /m +(1) HI/m = (I),

for the profile, where

(q)I /m

pg(m+r)hcZ+I/m=A 2" (2m+r) [I + I/m,

and A and m are constants appearing in the basal sliding law

u = u(r) = {crr~ (.<: = o)jA}m.

q is the constant accumulation, p is the uniform density of ice, he is the height at the centre of the ice sheet, and [ the radius .

As for the plane-flow profile, the condition of small surface slope is violated in the axi­symmetric profile except in some central zone. Nye's analysis is independent of the flow law and any temperature dependence. He has, nevertheless, deduced that observed temperature profiles give rise to an approximately uniform horizontal velocity profile above a thin bottom layer, and used this as the basic kinematic approximation. The analysis in this paper does not include any temperature variation so that Nye's deduction cannot be tested, nor can any effect of temperature variation be predicted.

Following the approach of M and J, for the axisymmetric problem the boundary conditions are

z = 0: (Jr~ = >.. (h)(u)I /m, (2)

Z = 0: w = -b(h) ~ 0, (3) where the basal drainage b and the sliding coefficient>.. depend on the overlying height of ice. If q is the accumulation (volume flux per unit horizontal area) referred to the horizontal cross-section, then

z = h(r) : w-h'u = -q(h).

The free-surface conditions are

(r+y2)(crrr+(J~~)+(r-y2) ( cr~~-(Jrr) + 4ycrrz = 0,

and symmetry requires

r = 0: u = 0, (Jrz = 0, h' = y = o.

(5) (6)

Formulation of the full problem in dimensionless variables introduces a small parameter v. However, the approximation v = 0 does not permit a bounded ice sheet. Rescaling the hori­zontal coordinate by a small factor £(v) ~ r allows a regular perturbation solution in £ .

v and £ are identical to the parameters which arose in the plane-flow analysis, and again £

STEADY PROFILE OFAXISYMMETRIC ICE SHEET

defines the small magnitude of the surface slope. The small-slope solution is valid up to the margin (h = 0) provided the sliding coefficient )' (h) ~ >"oh as h -)0 0, as for the plane flow case.

The results obtained are compared with those for the plane-flow solution (M and ] ) for the case m = 1. In all illustrations it is found that the aspect ratio decreases (or is unchanged), as predicted by Nye (1959), but here the results demonstrate the dependence of the decrease on the accumulation rate and the sliding law.

2. BALANCE LAWS AND CONSTITUTIVE EQUATIONS

The ice is assumed to be incompressible, so mass balance requires

OW 0 r 0.0+ or (ru) = o.

Inertia terms are negligible in this slow viscous flow so momentum balance requires

OCIrr OCIrz CIyy-CIoo } -+-+ =0 or 0.0 r '

OCIrz OCIzz CIyZ -+-+ --pg = o.

or 0.0 r

(8)

(9)

Following M and ], the ice is assumed to be an incompressible non-linear viscous fluid with a temperature-dependent rate factor (Morland, 1979), so

a+pl -- = 4>1 (lz, 13) fi +4>z(1z, 13)[DZ-i1zl ],

CIo

where

p = -1 tr a, lz = t tr fiz, 13 = det fi, and D = D /Doa(T). The non-zero components of D are

ou Dry = or'

u Doo = -,

r Dzz = 0.0' OW }

I ( ou Ow) Drz = Dzr = 2" 0.0 +Tr '

( 10)

(11)

T denotes temperature, and a( T) is the rate factor, normalized by a( To) = I for some temperature T o, with a' (T) ~ o. CIo and Do denote a constant stress magnitude and a constant strain-rate magnitude respectively, so that fi, the invariants 1z, 13, and the response functions 4>1' 4>z are dimensionless.

The more commonly adopted laws for D give the simpler form

I (a+PI)z Jz = -tr -- , 2 CIo

which implies

l z = JzwZ (]z) = G(Jz) say,

4>1 = 4>1 (1z), (14)

With

CIO = 105 N m-z, To = 273 K, ( 15)

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Glen's law gives

W(]Z) = I.5k(3]z)<n-I)/z, <PI (l z) = ik-I/n(!lz)-<n-I)/zn,}

k=0.17, n= I.3--r4,

and the Colbeck and Evans polynomial law gives

w(]z) = I.5 (Co+3CI]Z+gC2]z2), }

Co = 0.21, Cl = 0.14, C2 = 0.055· (17)

In the following analysis it is assumed that in the general law ( IQ) the stress contribution from a non-zero <P2 term is not of greater magnitude than that of the <PI term. It is also assumed that 1<P21 ~ 0 (1), but the singularity in <PI arising from Glen's law (Equation (16» will be considered.

3. DIMENSION LESS FORMULATION AND THE SMALL PARAMETER

Introduce dimensionless variables by

(r, z, h, l) = ho(R, Z, H, L), (a, p) = pghoCE., P), }

(u, w, q, b) = qm (U, W, Q, B),

where ho is a magnitude of the maximum ice thickness and qm is a magnitude of maximum accumulation density. Take qm = q(ho) on the assumption that ablation at lower heights does not significantly exceed this value. It is supposed that the drainage magnitude is not greater than that of Q. Define

h'(r) = H'(R) = r(H).

Setting

in terms of a dimensionless stream function 'I'" (r, z), satisfies the mass-balance equation (8). The momentum equations (9) become

o~rr o~rz ~rr-~/J/J _ } oR + oZ + R - 0,

(21) o~rz o~zz ~rz oR + oZ + }f- 1 = 0.

Boundary conditions (2)-(7) become, for z = 0:

for z = H(R):

[

I O'¥ ] I/m Lrz ~ A(H) R oZ '

)"'(h) qmI/m

A(H) = h ' pg 0

1 0'1'" R oR = B(H);

1 Cl'¥ r 0'1'" R ClR+R oZ = Q (H),

( 1 + rz) ( ~rr+ ~zz) + (I - rz)(~zz- Lrr) + 4r~rz = 0,

-(I -rz)~rz+r(~zz-~rr) = 0;

STEADY PROFILE OFAXISYMMETRIC ICE SHEET

and for R = 0:

I 00/ U = 0 ~ lim - - = 0

R-+o R oZ ' r = o.

The tensors D, Dz have non-zero components

b rr = 0 o~ (i ~;) , Doo = 0 ~z ~~, b zz = - 0 o~ (~ ~~) , o { 0 (I 0'Y) 0 (I 0'Y)} 0 {I ( OZo/ OZ'Y) I 0'Y}

b rz = ;- oZ R oZ - oR R oR = ;- R oZz - oRz + Rz oR '

[{ 0 (I o'Y)}Z I {o (I 0'Y ) 0 (I o'Y)}Z] (DZ)zz = OZ oZ R oR +4 oZ R oZ -oR R oR '

o = qm/aDohQ,

where, under typical conditions (M and ]),

IO- Z > 0 > 3 X 10-4. (29)

To deal with the singularity in 1>1 at l z = 0 for Glen's law (see M and ]), let

1>ID = ,p1 (Dl z- a), 0 :!( IX < t, (30) so that

is bounded, and for Glen's law

n-I I IX = -- < -

2n 2 '

(3 1 )

2 ( 3) (n-I) /zn ,pI = 3" 4 k- I/n = 0(1),

and,pl is const!'lnt, while the finite viscosity case is obtained by setting IX = 0, with 1>1 = 0(1). Hence Equations (30) and (31 ) cover both the cases with a power-law singularity and those which are non-singular. Equation (10) is now

~+PI = v[,pllz-aO+ol+za1>z{Dz-i1zl }], (33) where D Z = ozO and

The parameters 0, v, and s are identical to those of the plane-flow analysis and typically

v ~ I, (35)

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(see table I, M and ]), which is the condition adopted. Note also that

8 == 8/s ~ ()(l), for practical conditions.

By setting v == 0 in the momentum equations a uniform parallel slab of infinite extent is predicted with

P == H-Z, excluding a margin at finite radius.

4. SCALED VARIABLES AND SLOPE MAGNITUDE

H == constant,

For a non-zero surface slope, P, to leading order, must not be independent of R, so the horizontal momentum balance given by the first of Equations (2 I ) must involve the shear stress gradient in Z. This suggests a horizontal coordinate contraction

dH . r == EY == E-,

dp (38)

(cf. M and J), where p, y == O( I) and r has magnitude E. To retain the surface accumulation balance given by Equation (24), a stream-function scaling

.p == E2o/ == 0 (1),

is required, so Equation (24) becomes

Z==H: I 'O.p Y 'O.p P 'O p +p az == Q.

(39)

Note that for plane flow '¥ was scaled by E but the relations between the velocity components and the axisymmetric stream function are different.

As before, a balance in the momentum equations requires E ~ I , and only leading-order terms are presented for brevity. Thus, by Equations (28) and (33)

~rz ~ v [; ~IE2a-li-a a~ G :~)-~ C28I+2a~2i] .

The balance now requires

VE2a- 2 == 0 (1), and hence

E == v l /(2-2a) == vn/(n+l) == 88-n /(n+I).

This restriction is identical to that in the plane-flow case so the ~2 term does not contribute to I: to leading order with the assumption 1~21 ~O ( I ). For most, if not all, practical conditions

E ~ I,

which is assumed here.

STEADY PROFILE OFAXISYMMETRIC ICE SHEET

5. LEADING-ORDER APPROXIMATION

A power-series expansion in E is again appropriate, so let P = Po+O (E), tf = rfo+O (E), and

Yo = 7]' (p).

Denote all leading-order quantities by a subscript or superscript 0, so

~rro = ~ooo = ~zzo = -Po,

. [I 0 (I OrfO)] 2

lo = 2' oZ P oZ '

I Otfo W ----0- p op .

Proceeding with the solution as before gives rise to the ordinary differential equation

:p [7]P ( -jjt~)] m - ~g, (- 7]'7])-(:)2gZ(- 7]'7])) = p{Q(7])-B(7])}

= pQ*(7]),

for the free surface H = 7] (p), where

j = O(n- ,)(m+I) /zm(n+,) ::::;:; 0(1),

A is normalized on the scale El = E (n = I) and so is independent of n. The g functions arising from the constitutive laws are unaltered, and so for Glen's law and the Colbeck and Evans polynomial laws respectively

g(t) = 3 i (n+Ilkt n, }

g(t) = 3t(Co+3C,0t2+gCzOZt4),

and

I I

g,(t) = f g(t') dt', gz(t) = J g,(t') dt'. o o

Here the argument - 7]'7] is always positive. tfo is given by

where

32 JOURNAL OF GLACIOLOGY

The leading order stress components are

LrrO = LOll = ~zzo = -Po = -fY)(p)-Z}, }

Lrzo = -E7/(p){1](p)-Z},

and these have the same form as for plane flow.

6. GENERAL PROPERTIES AND VALIDITY

Equation (48) is analogous to equation (68) in M and ]. It differs in the appearance of the p term on both sides of the equation which leads to a stricter restriction on A ( 1]) and also requires a slightly different numerical approach. Rewrite Equation (48) as a first-order ordinary differential equation for r/ = Yo CY) ) :

YO:1) [1]P (_;£)m + 1]2pQ (- Y01) )] = pQ*, }

Q (X) ~ Qoxn as X -7 o.

(55)

For Glen's law and Colbeck and Evans' law respectively

Q = 3!(n+')k( - Yo1))n } n+ 2

Q = Co( -Yo1)) +!8C, ( -Yo1])3+ 2/8'C2( -Yo1))5.

(56)

Zero mass flux requires that, integrating over the surface and the bed, .R

f Q* dS = 0 => f Q*p dp = 0,

o

which implies for 1) -70:

Yo(1)) ~ -Ym1]P(1 -Y,1)+ ·· .), (58)

o ~ rnt < As 1) -'> 1)c = 1) (0), A and Q* are finite and Yo, P -7 0, so the essen tial behaviour of Equation (55) is

where l = min (rn, n) . Hence

Yo ~ (1]c-1))'/(!+Il, p :: (1)c-1))!/(!+Il. (60)

As 1) -70(0 ~ Z ~ 1)) or Yo -7 0 the dominant terms ofifJo, in both limits, from Equation (52), are

p

ifJo- f pB(1]) dp ~ p(_ Yo)n1]n+2 or (61 )

o

Thus, as Yo -7 0, 1) -7 1)c,

STEADY PROFILE OFAXISYMMETRIC ICE SHEET 33

d 'ffi .. f b . (6) 0 (I 0</;0) 02 (I 0</;0) Direct 1 erentlatlOn 0 oth terms m I show that bounded op pap' Op2 pap

requires (i) if l = n then l = n = I and m = I or 2 or m ~ 3; (ii) if l = m then l = m = I

and n = I or 2 or n ~ 3. These are the same restrictions as for plane flow. I t now remains to look at the behaviour of Yo as 7J -+ 0 to find any restrictions on t and m

to ensure that f3 ~ o. Let

Q*(7J) = -Qo(I-QI7J+ ... ) as 7J ~ 0, Qo > o. (63)

. f d7J Smce p = - then as 7J ~ 0, Yo

I [7J1

-P YI7J

2-

P] p ~ pm-- --+--Ym I-f3 2-f3

Substituting these asymptotic values in Equation (55) and balancing both sides of the equation leads to

m(t- I) f3 - = 0 ~ t = I, - I +m (65)

and

(66)

where pm = P (7J = 0). The restriction f3 = 0 is the same as for the plane-flow problem. The appearance of the P term in Equation (55) means that the problem cannot be con­

verted to an initial-value problem as before, but must be solved as a two-point boundary-value problem. Equation (55) can be written as two first-order ordinary differential equations for 7J, P with Yo as the independent variable:

where

d7J } dyo = F1 (p, 7J, YO)fF2 (P, 7J, Yo),

(67) dp I d7J ----dyo Yo dyo'

F1(p, 7J, Yo) = m7J ( - ;~)m -Yo7J3Q'( -Yo7J) ,

F 2 (p, 7J, Yo) = Q*-( - ;~) m [~+YO {I +m (1-7J~')}]_ -(;+2Y07J) Q( -Yo7J) +r027J2Q'( -Yo7J),

and the prime denotes the derivatives with respect to the argument. In the centre, as p, Yo -+ 0

(68)

since l = I. Also

(69)

34 JOURNAL OF GLACIOLOGY

where Ac = A (1] = 1]c), Qc = Q* (7) = 7)c) . The limit of Fz, Equation (69), is bounded for l ~ I in the light of the second of Equations (55) and Equation (60) . However, since l = I

the limit must be evaluated explicitly in order to determine dp/dyo at 7) = 7)c, P = Yo = o. As p, Yo ---+ 0 the essential behaviour of Equation (48) is

d [ {1]Cyo}m ] dp 1]cP - JAc + 1] c2pQ(- 1]cyo)+O(p3) = pQC+ O(p2),

which integrated gives

7)c [ 1]C Yo] m 7)c2

- -""""'\ +- Q ( -1]cYo) = tQc+O(p), p jllc p

and so

So finally, Equations (67) are integrated numerically using the explicit end-point values

d7) p = Yo = 0: - - 0 dyo - ,

7) = 0, Y = Ym:

m = n = I,

m = I, n> I,

m> I, n = I,

:;0 = Y~YI' }

dp I

dyo = - YmzYI '

using the first of Equations (58) and Equation (65), where Ym and YI are given by Equation (66), and Qoln_1 for Glen's law and Colbeck and Evans' law respectively is

The boundary conditions are:

1] = 0, p = Pm-unknown, }

Yo = 0: p = 0, 7) = 1]c-unknown.

Solutions for various values of the physical parameters were found using a shooting technique. It was found that numerical convergence hinged on good initial choices of pm and 7)c .

7. ILLUSTRATIONS

For comparison between laws with different exponents n, all results are expressed in terms of the same dimensionless horizontal coordinate p*, based on the scale factor El = E (n = I), and so, from Equation (44)

El p* = - p = (J(n-ll /z(n+I) p. E

STEADY PROFILE OFAXISYMMETRIC ICE SHEET 35

All calculations adopt the values in Equations (15) with a = 0.1, qm = 3 X 10-9 m S-l, so that () = 0.09. The real slope magnitude E for different nand ho is shown in table I, M and J. Illustrations are shown for linear A and Q*, namely

For comparison with Nye's (1959) profile, the latter is rewritten in terms of the dimension­less coordinates (p, 7] ), taking q = qm, A = A(ho) giving

where 7]c or Pm has to be prescribed. Comparisons between Nye's solution as given by Equation (79) and the complete small­

slope solution are shown in Figure 2 with Qo = Ao = I, where the value of 7]c in Equation (79) has been set to that obtained by the analysis presented here.

Figure 3 shows the profiles obtained for four sets of (Ao, Qo, m) using the Colbeck and Evans polynomial law. These profiles are analogous to those shown in figure 3, M and J, and it is clear that the effects of changing the parameters (Ao, Qo, m) are similar for the plane flow and axisymmetric cases. Values of Pm* and 7]c for (Ao, Qo) = 1,5, 10 with m = I, are shown in Table I (cf. M and J, table II ) .

It now remains to compare the plane-flow and axisymmetric solutions. Plane-flow analysis is simpler and other physical solutions, such as a non-symmetric ice sheet, may be considered. However, by looking at the axisymmetric case, some insight into the effect of a third dimension can be obtained. Nye (1959) predicted a decrease in the aspect ratio of

m= I,

m = 2.5.

3

·5

f

2 1·5 ·5 Fig. 2. Comparison of Nye's solution with the complete small-slope profile for Glen's law with linear A and Q*, with

>.0 = Qo = I; (i) -- m = I, n = I, (ii) ----- m = 2, n = 3.

1

2

JOURNAL OF GLACIOLOGY

1 /.

1/ 1 . 1/

/.. 7

/,>

1·5

" " "

/

/

'" '" '"

/

:

'" "

/ /

/

..

..

./

·5

---

_. _. ........ . .... ..

3

1·5

·5

Fig. 3. Comparison of profiles for Colbeck and Euans' law with linear A and Q*,. (i) --- Aa = 1, Qo = 1, m = 1,

(ii) ----- >-0 = E, Qo = 5, m = I, (iii) .. . ... Aa = 5, Qo = I , m = I , (iu) -.-.- . Aa = I , Qo = I, m = 3.

TABLE 1. RADIUS pm* AND HEIGHT 'le FOR LINEAR ACCUMULATION Q* (TJ) AND LINEAR SLIDING LAW (m = I ) WITH LINEAR COEFFICIENT A(TJ). SOLUTIONS FOR COLBECK

AND EVANS' LAW (CE) AND GLEN'S LAW WITH n = 1,3

Ao 5 10

Qo Law pm* 'le pm* 'le pm* 'le

CE 2.07 1 0.97 1 1.041 0.905 0.821 0.866 2·°57 0.976 1.01 3 0.9 18 0.785 0.879

3 1.991 1.000 0·9°5 0 ·993 0.661 0.978

CE 1.647 1.555 0.967 1.413 0.840 1.359 5 1.607 1.574 0.878 10432 0.729 1.365

3 1.519 1.649 0.804 1.552 0.669 1·5°1

CE 1.309 1.678 0.821 1.529 0·735 1.483 10 1.255 1.704 0·702 1·543 0.59 1 1.472

3 1.205 1.775 0.697 1.650 0.605 1.601

The analysis presented here and in M and J does not yield a simple formula for the change in the aspect ratio, but a comparison between table II (M and J) and Table I confirms a decrease (or no change) in all cases considered, as shown by Table II. These values range from 0 - 17 % compared with Nye's prediction of 2 I %. I t is, however, apparent that changes in the accumulation rate and the sliding law affect this decrease.

Finally, Figure 4 shows both the plane-flow and axisymmetric profiles on the same horizontal axis for the Colbeck and Evans polynomial law, with Qo = Ao = I, and m = I.

STEADY

,-'" ,-

,-( !"E,I .f )

,-

'" ,-

'" 2

PROFILE OF AXISYMMETRIC ICE SHEET

TABLE H. PERCENTAGE DECREASE IN THE ASPECT RATIO FROM PLANE TO AXI-

Qo

5

10

'" '" ,-,-

,-

1·5

SYMMETRIC FLOW

"" Law

CE

3 CE

3 CE

I 3

",

'" '"

3 2 0 6 5 2

8 6 4

5 10

8 I1 7 10

4 14 16 12 14 11 15

15 17 12 15 14 16

-- ------

·5

37

-----

·5

Fig. 4. Comparison between plane and axisymmetric jlowfor Colbeck and Evans' law with linear A, Q*, where Ao = Qo = I,

and m = I; (i) --- plane jlow, (ii) ----- axisymmetric jlow.

ACKNOWLEDGEMENT

I am indebted to Dr 1. W. Morland for helpful discussions and for reading of the initial draft and to the Science Research Council for the award of a research studentship.

MS. received 13 December 1979

REFERENCES

Morland, L. W. 1979. Constitutive laws for ice. Cold Regions Science and Technology, Vo!. I, No. 2, p. 101-08. Morland, L. W., and Johnson, 1. R. 1980. Steady motion of ice sheets. Journal qf Glaciology, Vo!. 25, No. 92,

p.229-46. Nye, J . F. 1959. The motion of ice sheets and glaciers. Journal of Glaciology, Vo!. 3, No. 26, p. 493-507. Weertman, J. 1961: Equilibrium profile of ice caps. Journal of Glaciology, Vo!. 3, No. 30, p. 953-64.